CN105653827A - Method for designing Terminal sliding mode controller of hypersonic vehicle - Google Patents

Abstract

The invention relates to a method for designing a Terminal sliding mode controller of a hypersonic vehicle. The Terminal sliding mode control has an advantage that a system on a sliding mode surface can be converged within limited time. In order to keep the nonlinearity of a model and to facilitate the design of a control system, an input-output linearization method is used for processing the model in the paper, after that, state variables are reselected according to the design requirements of the Terminal sliding mode control, and a nonlinear sliding mode surface is designed; a sliding mode control law is designed for the designed Terminal nonlinear sliding mode surface, to ensure that the system can reach the sliding mode surface, and to prove the stability of the system; finally, the given control law is subjected to simulation verification, and the results show that the designed sliding mode controller can realize the effective control of the hypersonic vehicle.

Description

Hypersonic aircraft Terminal sliding mode controller design method

Technical field

The present invention relates to hypersonic aircraft Terminal sliding mode controller design method, belong to aviation controller regulation technology field.

Background technology

Hypersonic aircraft is subject to the attention of countries in the world because of advantages such as its flight speed are fast, mobility strong, payload are big, becomes the emphasis of aircraft research and development of future generation. But owing to aircraft itself adopts fuselage engine integration design, cause that aircraft engine state and state of flight are mutually coupled; And the flight envelope of hypersonic aircraft is big, flight progress is complicated, its integral power model exists serious non-linear, structure is also very important with the uncertainty of parameter and external disturbance simultaneously, and this makes the controller design method of classics be difficult to design the controller meeting its performance requirement such as strong robustness, fast-response.

For the controller design problem of hypersonic aircraft, expanding both at home and abroad and study widely, the method adopted at present has H_��Optimum control, model reference adaptive, linear variation parameter, adaptive sliding-mode observer etc., the design of hypersonic aircraft controller has attracted the sight of a lot of scholar in the world.

The complete robustness of matching uncertainties and external disturbance etc. is received an acclaim by Sliding mode variable structure control because of itself. Longitudinal modelling adaptive sliding mode controller for hypersonic aircraft, demonstrate the effectiveness that sliding formwork controls, but the sliding-mode surface adopted is traditional linear sliding mode face, the exponential convergence in time of its state, convergence rate is difficult to meet aircraft when high-speed flight to the requirement responding rapidity.

Summary of the invention

Instruction Tracing Problem for hypersonic aircraft is studied by this method, space vehicle dynamic model is carried out linearisation by the theory first with I/O linearization, adopt Terminal sliding mode design method design con-trol device, designed sliding-mode surface is nonlinear, make the fast response time of system, can ensure that tracking error system arrives zero point in finite time, thus ensure that the quick tracking to command signal. System trajectory is after reaching the sliding mode stage to select linear sliding-mode surface to be able to ensure that, the motion of sliding mode be asymptotically stability or tracking error progressive converge to zero, and convergence rate is to be regulated by the parameter matrix of change sliding-mode surface, but, the status tracking error in linear sliding mode face can not converge to zero within the limited time.The control thought of Terminal sliding formwork, by being incorporated in the design of sliding-mode surface by nonlinear terms, breaches the limitation in conventional linear sliding mode face so that the sliding mode variable of system can converge to equilibrium point in " in the limited time ". Initial Terminal sliding mode is from Terminal attractor. Venkatar-manS.T. the design problem first analyzing Terminal sliding formwork is waited, and apply it in the middle of robot system, YuX. and man.Z.H. etc. are for SISO system and mimo system subsequently, and the control problem of Terminal sliding formwork has been carried out detailed research. Wu.Y.Q. wait and also utilize the method for recurrence to thoroughly discuss during Terminal sliding formwork controls the singular problem easily occurred. In this article, we will use for reference the thought design con-trol device of Terminal sliding formwork, and the status tracking error to ensure hypersonic aircraft can at Finite-time convergence. First the basic thought of Terminal is introduced.

In single order Terminal sliding formwork, the definition of single order Terminal sliding formwork is as follows:

$s=\stackrel{\·}{x}+{\mathrm{\βx}}^{q/p}=0---\left(1\right)$

Wherein x is a scalar, �� > 0, p, q (p > q), and p, q are positive odd number. No matter x is any real number, x^p/qSolution be necessary for real number.

System dynamic property on sliding-mode surface is

$\stackrel{\·}{x}=-{\mathrm{\βx}}^{q/p}---\left(2\right)$

Given arbitrary original state x (0) �� 0, then system will at Finite-time convergence to initial point. Solving equation (2)

$\frac{p}{p-q}\[x{\left(t^s\right)}^{(p-q)/p}-x{\left(0\right)}^{(p-q)/p}\]=-{\mathrm{\βt}}^s---\left(3\right)$

The time that system experiences can be obtained from state x (0) to initial point

$t^s=\frac{p}{\mathrm{\β}(p-q)}{\left|x\left(0\right)\right|}^{(p-q)/p}---\left(4\right)$

The explanation of Terminal limited mechanism is considered below, it is considered to the Jacobian matrix at equilibrium point x=0 place is

$J=\frac{\∂\stackrel{\·}{x}}{\∂x}=-\frac{\mathrm{\β}q}{{\mathrm{px}}^{(p-q)/p}}---\left(5\right)$

J is regarded as the eigenvalue �� of first order matrix, then when x �� 0⁺Time, J ��-��, then the track of system, naturally can with infinitely-great speed convergence to equilibrium point under the driving of the eigenvalue of minus infinity. Therefore system will at Finite-time convergence to initial point.

For nonlinear system, adopt the I/O linearization method based on differential geometric theory, the complexity reducing system controller design on the basis of mission nonlinear feature can retained, therefore it is contemplated that adopt I/O linearization method that system is processed. The party ratio juris is briefly described below.

For affine nonlinear system

$\begin{array}{c}\stackrel{\·}{x}=f\left(x\right)+G\left(x\right)u,G\left(x\right)=\[g_1\left(x\right),g_2\left(x\right),...,g_m\left(x\right)\]\\ y=h\left(x\right)\end{array}---\left(6\right)$

Wherein f (x), g (x) are smooth function.

First passing through respectively to function f (x), g (x) asks for the Lie derivatives about output function h (x) system, seeks the Relative order of system, and concrete form is:

$\begin{array}{c}{L}_f{h}_i=\frac{\∂h_i}{\∂x}f\\ L_f^k{h}_i=L_f\left(L_f^{k-1}{h}_i\right)\\ L_{gi}{h}_i=\frac{\∂h_i}{\∂x}{g}_i\end{array}---\left(7\right)$

IfThe Relative order then claiming system is r.

If the Relative order r=n of system, (n is systematic education), then system is to fully input linearization. Selection differomorphism converts

$\begin{array}{c}{\mathrm{\ζ}}_1=L_{f}h\left(x\right)\\ {\mathrm{\ζ}}_2=L_f^{2}h\left(x\right)\\ .\\ .\\ .\\ {\mathrm{\ζ}}_{n-1}=L_f^{r-1}h\left(x\right)\end{array}---\left(8\right)$

To (8) formula derivation, the state equation that can obtain transformation system is:

$\begin{array}{c}{\stackrel{\·}{\mathrm{\ζ}}}_1=L_f^{2}h\left(x\right)={\mathrm{\ζ}}_2\\ {\stackrel{\·}{\mathrm{\ζ}}}_2=L_f^{3}h\left(x\right)={\mathrm{\ζ}}_3\\ .\\ .\\ .\\ {\stackrel{\·}{\mathrm{\ζ}}}_{n-1}=L_f^{r-1}h\left(x\right)={\mathrm{\ζ}}_n\\ {\stackrel{\·}{\mathrm{\ζ}}}_n=L_f^{r}h\left(x\right)+L_g^{r-1}h\left(x\right)u\end{array}---\left(9\right)$

Observation equation (4), it is possible to find except finally with respect to ��_nEquation outside, all the other n-1 equations have been linear forms, and without controlled quentity controlled variable. Only with respect to ��_nState equation be nonlinear, but to input u, equation form is linear. Dirty [state] equation (9)

$\begin{array}{c}{\stackrel{\·}{\mathrm{\ζ}}}_1=L_f^{2}h\left(x\right)={\mathrm{\ζ}}_2\\ {\stackrel{\·}{\mathrm{\ζ}}}_2=L_f^{3}h\left(x\right)={\mathrm{\ζ}}_3\\ .\\ .\\ .\\ {\stackrel{\·}{\mathrm{\ζ}}}_{n-1}=L_f^{r-1}h\left(x\right)={\mathrm{\ζ}}_n\\ {\stackrel{\·}{\mathrm{\ζ}}}_n=v\end{array}---\left(10\right)$

Wherein

$v=L_f^{r}h\left(x\right)+L_g^{r-1}h\left(x\right)u---\left(11\right)$

Then state equation can become:

$\stackrel{\·}{\mathrm{\ζ}}=A\mathrm{\ζ}+Bv---\left(12\right)$

WhereinAt this moment, system form becomes linear, and remain the nonlinear characteristic of system so that system more easily processes.

This method adopts the hypersonic aircraft longitudinal direction model that NASA langley laboratory is announced to study, and its model is as follows:

Model hypothesis

(1) hypersonic aircraft is desirable rigid body, is namely left out the elastic free degree of wing etc.;

(2) centroid position, rotary inertia is the function of quality, and centroid position changes at the axis longitudinal axis all the time;

(3) aircraft center and reference centre of moment are on body X-axis;

(4) assume that aircraft layout is symmetrical, namely product of inertia I_xy, I_xz, I_yzPerseverance is zero;

(5) rotary inertia and the motor power established angle of control surface are ignored.

$\stackrel{\·}{V}=(Tcos\mathrm{\α}-D)/m-\mathrm{\μ}sin\mathrm{\γ}/r^2$

$\stackrel{\·}{\mathrm{\γ}}=(L+Tsin\mathrm{\α})/mV-(\mathrm{\μ}-V^{2}r)cos\mathrm{\γ}/{\mathrm{Vr}}^2$

$\stackrel{\·}{h}=Vsin\mathrm{\γ}$

$\stackrel{\·}{\mathrm{\α}}=q-\stackrel{\·}{\mathrm{\γ}}$

$\stackrel{\·}{q}=Myy/Iyy---\left(13\right)$

In formula, L is lift, and D is resistance, T motor power, and Myy is rolling moment, and Iyy is aircraft own rotation inertia, and r is the distance of aircraft and the earth's core, and the expression formula that each parameter is concrete is as follows:

$L=\frac{1}{2}{\mathrm{\ρV}}^2{\mathrm{SC}}_L$

$D=\frac{1}{2}{\mathrm{\ρV}}^2{\mathrm{SC}}_D$

$T=\frac{1}{2}{\mathrm{\ρV}}^2{\mathrm{SC}}_T$

$Myy=\frac{1}{2}{\mathrm{\ρV}}^{2}S\stackrel{\‾}{c}\[C_M\left(\mathrm{\α}\right)+C_M\left({\mathrm{\δ}}_e\right)+C_M\left(q\right)\]$

R=h+R_E(14)

In formula, �� is atmospheric density, and S is that electromotor effective cross section is amassed, and the expression formula of each coefficient is as follows:

C_L=0.6203 ��

C_D=0.6450 ��²+0.0043378��+0.003772

$C_T=\left\{\begin{array}{cc}0.02576\mathrm{\&beta;}& \mathrm{\&beta;}<1\\ 0.0224+0.00336\mathrm{\&beta;}& \mathrm{\&beta;}>1\end{array}\right.$

C_M(��)=-0.035 ��²+0.036617��+5.3216��10^-6

C_M(��_e)=ce (��_e-��)

$C_M\left(q\right)=(\stackrel{\&OverBar;}{c}/2V)q(-6.796{\mathrm{\&alpha;}}^2+0.3015\mathrm{\&alpha;}-0.2289)---\left(15\right)$

�� in formula_eRepresenting elevator drift angle, �� represents electromotor mode, and its expression formula is:

$\stackrel{\&CenterDot;\&CenterDot;}{\mathrm{\&beta;}}=-2{\mathrm{\&xi;\&omega;}}_n\stackrel{\&CenterDot;}{\mathrm{\&beta;}}-{{\mathrm{\&omega;}}_n}^2\mathrm{\&beta;}+{{\mathrm{\&omega;}}_n}^2{\mathrm{\&beta;}}_c---\left(16\right)$

��_cInput is controlled for engine throttle.

Reality according to hypersonic aircraft, controls input �� by engine throttle_cWith elevator drift angle ��_eAs controlling input, speed V and height H is elected in output as.

According to (13)-(16) formula it is found that the kinetic model of hypersonic aircraft exists serious non-linear and close coupling, and not aobvious containing input in equation, it is impossible to directly design con-trol device, it is necessary to model is converted. I/O linearization method is the important method of Nonlinear Control System Design and process, make system form shows linear behavio(u)r by system is differentiated, and itself also retain the nonlinear characteristic of this system, therefore here adopt I/O linearization method to be converted by model, be then controlled device design.

Select state variable x=[V �� �� �� h]^T, control input u=[��_c��_e]^T, definition system is output as y=[V, h]^T, according to I/O linearization method, respectively to speed V with highly carry out 3 times and 4 subdifferentials, it is possible to obtain:

$\stackrel{\&CenterDot;}{V}=(Tcos\mathrm{\&alpha;}-D)/m-\mathrm{\&mu;}sin\mathrm{\&gamma;}/r^2$

$\stackrel{\&CenterDot;\&CenterDot;}{V}={\mathrm{\&omega;}}_1\stackrel{\&CenterDot;}{x}/m$

$\stackrel{\&CenterDot;\&CenterDot;\&CenterDot;}{V}=({\mathrm{\&omega;}}_1\stackrel{\&CenterDot;\&CenterDot;}{x}+{\stackrel{\&CenterDot;}{x}}^T{\mathrm{\&Omega;}}_2\stackrel{\&CenterDot;}{x})/m---\left(17\right)$

$\stackrel{\&CenterDot;}{h}=Vsin\mathrm{\&gamma;}$

$\stackrel{\&CenterDot;\&CenterDot;}{h}=\stackrel{\&CenterDot;}{V}sin\mathrm{\&gamma;}+V\stackrel{\&CenterDot;}{\mathrm{\&gamma;}}cos\mathrm{\&gamma;}$

$\stackrel{\&CenterDot;\&CenterDot;\&CenterDot;}{h}=\stackrel{\&CenterDot;\&CenterDot;}{V}sin\mathrm{\&gamma;}+2\stackrel{\&CenterDot;}{V}\stackrel{\&CenterDot;}{\mathrm{\&gamma;}}cos\mathrm{\&gamma;}-V{\stackrel{\&CenterDot;}{\mathrm{\&gamma;}}}^{2}sin\mathrm{\&gamma;}+V\stackrel{\&CenterDot;\&CenterDot;}{\mathrm{\&gamma;}}cos\mathrm{\&gamma;}$

$\begin{array}{c}{h}^{\left(4\right)}=\stackrel{\&CenterDot;\&CenterDot;\&CenterDot;}{V}\sin \mathrm{\&gamma;}+3\stackrel{\&CenterDot;\&CenterDot;}{V}\stackrel{\&CenterDot;}{\mathrm{\&gamma;}}\cos \mathrm{\&gamma;}-3\stackrel{\&CenterDot;}{V}{\stackrel{\&CenterDot;}{\mathrm{\&gamma;}}}^2\sin \mathrm{\&gamma;}+\\ 3\stackrel{\&CenterDot;}{V}\stackrel{\&CenterDot;\&CenterDot;}{\mathrm{\&gamma;}}\cos \mathrm{\&gamma;}-3V\stackrel{\&CenterDot;}{\mathrm{\&gamma;}}\stackrel{\&CenterDot;\&CenterDot;}{\mathrm{\&gamma;}}\sin \mathrm{\&gamma;}-V{\stackrel{\&CenterDot;}{\mathrm{\&gamma;}}}^3\cos \mathrm{\&lambda;}+V\stackrel{\&CenterDot;\&CenterDot;\&CenterDot;}{\mathrm{\&gamma;}}\cos \mathrm{\&gamma;}\end{array}---\left(18\right)$

WhereinFor the ease of isolating controlled quentity controlled variable, selectWherein After differential, controlling input and be already present in the differential equation, output kinetics equation can be written as:

$\left[\begin{array}{c}\stackrel{\&CenterDot;\&CenterDot;\&CenterDot;}{V}\\ h^{\left(4\right)}\end{array}\right]=\left[\begin{array}{c}{f}_V\\ f_h\end{array}\right]+\left[\begin{array}{cc}{b}_{11}& b_{12}\\ b_{21}& b_{22}\end{array}\right]u---\left(19\right)$

In formula

$f_V=({\mathrm{\&omega;}}_1{\stackrel{\&CenterDot;\&CenterDot;}{x}}_0+{\stackrel{\&CenterDot;}{x}}^T\mathrm{\&Omega;}\stackrel{\&CenterDot;}{x})/m$

$\begin{array}{c}{f}_h=3\stackrel{\&CenterDot;\&CenterDot;}{V}\stackrel{\&CenterDot;}{\mathrm{\&gamma;}}\cos \mathrm{\&gamma;}-3\stackrel{\&CenterDot;}{V}{\stackrel{\&CenterDot;}{\mathrm{\&gamma;}}}^2\sin \mathrm{\&gamma;}+3\stackrel{\&CenterDot;}{V}\stackrel{\&CenterDot;\&CenterDot;}{\mathrm{\&gamma;}}\cos \mathrm{\&gamma;}\\ -3V\stackrel{\&CenterDot;}{\mathrm{\&gamma;}}\stackrel{\&CenterDot;\&CenterDot;}{\mathrm{\&gamma;}}\sin \mathrm{\&gamma;}-V{\stackrel{\&CenterDot;}{\mathrm{\&gamma;}}}^3\cos \mathrm{\&gamma;}+V\stackrel{\&CenterDot;\&CenterDot;\&CenterDot;}{\mathrm{\&gamma;}}\cos \mathrm{\&gamma;}+f_V\end{array}$

$b_{11}=\frac{{\mathrm{\&rho;V}}^2{\mathrm{Sc}}_{\mathrm{\&beta;}}{w}_n^2}{2m}cos\left(\mathrm{\&alpha;}\right)$

$b_{12}=-\frac{c_e{\mathrm{\&rho;V}}^{2}S\stackrel{\&OverBar;}{c}}{2mIyy}(Tsin\mathrm{\&alpha;}+\frac{\&part;D}{\&part;\mathrm{\&alpha;}})$

$b_{21}=\frac{{\mathrm{\&rho;V}}^2{\mathrm{Sc}}_{\mathrm{\&beta;}}{w}_n^2}{2m}sin(\mathrm{\&alpha;}+\mathrm{\&gamma;})$

$b_{22}=\frac{c_e{\mathrm{\&rho;V}}^{2}S\stackrel{\&OverBar;}{c}}{2mIyy}\left(Tcos\right(\mathrm{\&alpha;}+\mathrm{\&gamma;})+\frac{\&part;L}{\&part;\mathrm{\&alpha;}}cos\mathrm{\&gamma;}-\frac{\&part;D}{\&part;\mathrm{\&alpha;}}sin\mathrm{\&gamma;})$

${{\mathrm{\&omega;}}_1}^T=\left[\begin{array}{c}\left(\frac{\&part;T}{\&part;V}\right)cos\mathrm{\&alpha;}-\frac{\&part;D}{\&part;V}\\ \frac{-m\mathrm{\&mu;}cos\mathrm{\&gamma;}}{r^2}\\ -Tsin\mathrm{\&alpha;}-\frac{\&part;D}{\&part;\mathrm{\&alpha;}}\\ \left(\frac{\&part;T}{\&part;\mathrm{\&beta;}}\right)cos\mathrm{\&alpha;}\\ \frac{2m\mathrm{\&mu;}sin\mathrm{\&gamma;}}{r^3}\end{array}\right]$

${{\mathrm{\&pi;}}_1}^T=\left[\begin{array}{c}\frac{\&part;L/\&part;V+(\&part;T/\&part;V)\sin \mathrm{\&alpha;}}{mV}-\frac{L+T\sin \mathrm{\&alpha;}}{{\mathrm{mV}}^2}+\frac{\mathrm{\&mu;}\cos \mathrm{\&alpha;}}{V^2{r}^2}+\frac{\cos \mathrm{\&gamma;}}{r}\\ \frac{\mathrm{\&mu;}\sin \mathrm{\&gamma;}}{{\mathrm{Vr}}^2}-\frac{V\sin \mathrm{\&gamma;}}{r}\\ \frac{\&part;L/\&part;\mathrm{\&alpha;}+T\cos \mathrm{\&alpha;}}{mV}\\ \frac{(\&part;T/\&part;\mathrm{\&beta;})\sin \mathrm{\&alpha;}}{mV}\\ \frac{2\mathrm{\&mu;}\cos \mathrm{\&gamma;}}{{\mathrm{Vr}}^3}-\frac{V\cos \mathrm{\&gamma;}}{r^2}\end{array}\right]$

��₂=[��₂₁��₂₂��₂₃��₂₄��₂₅]

${\mathrm{\&omega;}}_{21}=\left[\begin{array}{c}\left(\frac{{\&part;}^{2}T}{\&part;V^2}\right)cos\mathrm{\&alpha;}-\frac{{\&part;}^{2}D}{\&part;V^2}\\ -\left(\frac{\&part;T}{\&part;V}\right)sin\mathrm{\&alpha;}-\frac{{\&part;}^{2}D}{\&part;V\&part;\mathrm{\&alpha;}}\\ \left(\frac{{\&part;}^{2}T}{\&part;V\&part;\mathrm{\&beta;}}\right)\cos \mathrm{\&alpha;}\\ 0\end{array}\right],{\mathrm{\&omega;}}_{22}=\left[\begin{array}{c}0\\ \frac{m\mathrm{\&mu;}sin\mathrm{\&gamma;}}{r^2}\\ 0\\ 0\\ \frac{2m\mathrm{\&mu;}cos\mathrm{\&gamma;}}{r^3}\end{array}\right]$

${\mathrm{\&omega;}}_{23}=\left[\begin{array}{c}0\\ \frac{m\mathrm{\&mu;}\sin \mathrm{\&gamma;}}{r^2}\\ 0\\ 0\\ \frac{2m\mathrm{\&mu;}\cos \mathrm{\&gamma;}}{r^3}\end{array}\right],{\mathrm{\&omega;}}_{24}\left[\begin{array}{c}\left(\frac{{\&part;}^2}{\&part;V\&part;\mathrm{\&beta;}}\right)\cos \mathrm{\&alpha;}\\ 0\\ -\left(\frac{\&part;T}{\&part;\mathrm{\&beta;}}\right)\sin \mathrm{\&alpha;}\\ 0\\ 0\end{array}\right]$

${\mathrm{\&omega;}}_{25}=\left[\begin{array}{c}0\\ \frac{2m\mathrm{\&mu;}cos\mathrm{\&gamma;}}{r^3}\\ 0\\ 0\\ \frac{-6m\mathrm{\&mu;}cos\mathrm{\&gamma;}}{r^4}\end{array}\right]$

��₂=[��₂₁��₂₂��₂₃��₂₄��₂₅]

${\mathrm{\&pi;}}_{21}=\left[\begin{array}{c}\frac{{\&part;}^{2}L/\&part;V^2+({\&part;}^{2}T/\&part;V^2)\sin \mathrm{\&alpha;}}{mV}-\frac{2\&lsqb;\&part;L/\&part;V+(\&part;T/\&part;V)\sin \mathrm{\&alpha;}\&rsqb;}{{\mathrm{mV}}^2}+\frac{2(L+T\sin \mathrm{\&alpha;})}{{\mathrm{mV}}^3}-\frac{2\mathrm{\&mu;}\cos \mathrm{\&gamma;}}{V^3{r}^2}\\ -\frac{\mathrm{\&mu;}\sin \mathrm{\&gamma;}}{V^2{r}^2}-\frac{\sin \mathrm{\&gamma;}}{r}\\ \frac{({\&part;}^{2}L/\&part;\mathrm{\&alpha;}\&part;V)+(\&part;T/\&part;V)\cos \mathrm{\&alpha;}}{mV}-\frac{\&part;L/\&part;\mathrm{\&alpha;}+T\cos \mathrm{\&alpha;}}{{\mathrm{mV}}^2}\\ \frac{({\&part;}^{2}T/\&part;\mathrm{\&beta;}\&part;V)\sin \mathrm{\&alpha;}}{mV}-\frac{(\&part;T/\&part;\mathrm{\&beta;})\sin \mathrm{\&alpha;}}{{\mathrm{mV}}^2}\\ -\frac{2\mathrm{\&mu;}\cos \mathrm{\&gamma;}}{V^2{r}^3}-\frac{\cos \mathrm{\&gamma;}}{r^2}\end{array}\right]$

${\mathrm{\&pi;}}_{22}=\left[\begin{array}{c}-\frac{\mathrm{\&mu;}\sin \mathrm{\&gamma;}}{V^2{r}^2}-\frac{\sin \mathrm{\&gamma;}}{r}\\ \frac{\mathrm{\&mu;}\cos \mathrm{\&gamma;}}{{\mathrm{Vr}}^2}-\frac{V\cos \mathrm{\&gamma;}}{r}\\ 0\\ 0\\ -\frac{2\mathrm{\&mu;}\sin \mathrm{\&gamma;}}{{\mathrm{Vr}}^3}+\frac{V\sin \mathrm{\&gamma;}}{r^2}\end{array}\right],{\mathrm{\&pi;}}_{23}=\left[\begin{array}{c}\frac{({\&part;}^{2}L/\&part;V\&part;\mathrm{\&alpha;})+(\&part;T/\&part;V)\cos \mathrm{\&alpha;}}{mV}-\frac{\&part;L/\&part;\mathrm{\&alpha;}+T\cos \mathrm{\&alpha;}}{{\mathrm{mV}}^2}\\ 0\\ \frac{{\&part;}^{2}L/\&part;{\mathrm{\&alpha;}}^2-T\sin \mathrm{\&alpha;}}{mV}\\ \frac{(\&part;T/\&part;\mathrm{\&beta;})\cos \mathrm{\&alpha;}}{mV}\\ 0\end{array}\right]$

${\mathrm{\&pi;}}_{24}=\left[\begin{array}{c}\frac{({\&part;}^{2}T/\&part;V\&part;\mathrm{\&beta;})sin\mathrm{\&alpha;}}{mV}-\frac{(\&part;T/\&part;\mathrm{\&beta;})sin\mathrm{\&alpha;}}{{\mathrm{mV}}^2}\\ 0\\ \frac{(\&part;T/\&part;\mathrm{\&beta;})cos\mathrm{\&alpha;}}{mV}\\ 0\\ 0\end{array}\right],{\mathrm{\&pi;}}_{25}=\left[\begin{array}{c}-\frac{2\mathrm{\&mu;}cos\mathrm{\&gamma;}}{V^2{r}^3}-\frac{cos\mathrm{\&gamma;}}{r^2}\\ -\frac{2\mathrm{\&mu;}\sin \mathrm{\&gamma;}}{{\mathrm{Vr}}^3}+\frac{V\sin \mathrm{\&gamma;}}{r^2}\\ 0\\ 0\\ -\frac{6\mathrm{\&mu;}\cos \mathrm{\&gamma;}}{{\mathrm{Vr}}^4}+\frac{2V\cos \mathrm{\&gamma;}}{r^3}\end{array}\right]$

Converting through I/O linearization, the nonlinear model of hypersonic aircraft has been converted into pro forma linear model, and the nonlinear characteristic of model itself have also been obtained maximum reservation simultaneously.

The control target of hypersonic aircraft is to control one given command signal of aircraft output tracking, and ensures stablizing of system itself. Given command signal is y_com=[V_d(t),h_d(t)]^T, then following the tracks of target can be expressed as:The tracking error of definition system is:The target then controlled is the tracking error of guarantee systemBut the flight speed of hypersonic aircraft is fast, the relative response time requirement to system is also high, the controller of traditional exponential convergence is difficult to the requirement meeting aircraft to controlling system finite time convergence control, therefore the controller controlling to design finite time convergence control with Terminal sliding formwork is here considered, it is ensured that the Fast Convergent of system.

According to the thinking that Terminal sliding formwork controls, the sliding-mode surface of design tracking error is as follows:

$\begin{array}{c}{S}_V={\stackrel{\&CenterDot;\&CenterDot;}{e}}_V+{\mathrm{\&beta;}}_1{{\stackrel{\&CenterDot;}{e}}_V}^{q_0/p_0}+{\mathrm{\&beta;}}_2{e_V}^{q_1/p_1}\\ S_h={\stackrel{\&CenterDot;\&CenterDot;\&CenterDot;}{e}}_h+{\mathrm{\&alpha;}}_1{{\stackrel{\&CenterDot;\&CenterDot;}{e}}_h}^{q_0/p_0}+{\mathrm{\&alpha;}}_2{{\stackrel{\&CenterDot;}{e}}_h}^{q_1/p_1}+{\mathrm{\&alpha;}}_3{e_h}^{q_2/p_2}\end{array}---\left(20\right)$

HereIn like manner can push awayWithDesigned sliding-mode surfaceQi and pi (i=0,1,2) is positive odd number, and pi > qi. ��₁, ��₂, ��₁, ��₂And ��₃For arithmetic number.

According to the condition that sliding formwork controls, when system arrives sliding-mode surface, S=0, namely

${\stackrel{\&CenterDot;\&CenterDot;}{e}}_V+{\mathrm{\&beta;}}_1{{\stackrel{\&CenterDot;}{e}}_V}^{q/p}+{\mathrm{\&beta;}}_2{e_V}^{q/p}=0$

${\stackrel{\&CenterDot;\&CenterDot;\&CenterDot;}{e}}_h+{\mathrm{\&alpha;}}_1{{\stackrel{\&CenterDot;\&CenterDot;}{e}}_h}^{q/p}+{\mathrm{\&alpha;}}_2{{\stackrel{\&CenterDot;}{e}}_h}^{q_1/p_1}+{\mathrm{\&alpha;}}_3{e_h}^{q_2/p_2}=0---\left(21\right)$

Observation type (22) it is found thatIt is an equilibrium point of equation (22), and on sliding-mode surface, system equation convergence rate is power function convergence, so system can at Finite-time convergence to initial point.AndWhen, hypersonic aircraft also completes the effective tracking to command signal. So the sliding-mode surface of design meets the requirement of finite time convergence control.

Below for the sliding-mode surface of design, design corresponding sliding formwork control law, to ensure reachability and the global stability of System with Sliding Mode Controller.

Definition Lyapunov function is:

$V=\frac{1}{2}{S}^{T}S---\left(22\right)$

Above formula is differentiated:

$\begin{array}{c}\stackrel{\&CenterDot;}{V}=S^T\stackrel{\&CenterDot;}{S}\\ =S^T\left[\begin{array}{c}{\stackrel{\&CenterDot;\&CenterDot;\&CenterDot;}{e}}_V+{\mathrm{\&beta;}}_{1}q/p{\stackrel{\&CenterDot;\&CenterDot;}{e}}_V^{\frac{q-1}{p}}+{\mathrm{\&beta;}}_2{q}_{2}p{{\stackrel{\&CenterDot;}{e}}_V}^{\frac{q_1-1}{p_1}}\\ {e^{\left(4\right)}}_h+{\mathrm{\&alpha;}}_{1}q/p{{\stackrel{\&CenterDot;\&CenterDot;\&CenterDot;}{e}}_h}^{\frac{q-1}{p}}+{\mathrm{\&alpha;}}_2{q}_2/p_2{{\stackrel{\&CenterDot;\&CenterDot;}{e}}_h}^{\frac{q_1-1}{p_1}}+{\mathrm{\&alpha;}}_3{q}_2/p_2{{\stackrel{\&CenterDot;}{e}}_h}^{\frac{q_2-1}{p_2}}\end{array}\right]\end{array}$

Comparison formula (20), can obtain

$\stackrel{\&CenterDot;}{V}=S^T\left\{\left[\begin{array}{c}{f}_V-\stackrel{\&CenterDot;\&CenterDot;\&CenterDot;}{V}+{\mathrm{\&beta;}}_{1}q/{\stackrel{\&CenterDot;\&CenterDot;}{e}}_V^{\frac{q-1}{p}}+{\mathrm{\&beta;}}_2{q}_2/p_2{{\stackrel{\&CenterDot;}{e}}_V}^{\frac{q_1-1}{p_1}}\\ f_h-{h^{\left(4\right)}}_d+{\mathrm{\&alpha;}}_{1}q/p{{\stackrel{\&CenterDot;\&CenterDot;\&CenterDot;}{e}}_h}^{\frac{q-1}{p}}+{\mathrm{\&alpha;}}_2{q}_2/p_2{{\stackrel{\&CenterDot;\&CenterDot;}{e}}_h}^{\frac{q_1-1}{p_1}}+{\mathrm{\&alpha;}}_3{q}_2/p_2{{\stackrel{\&CenterDot;}{e}}_h}^{\frac{q_2-1}{p_2}}\end{array}\right]+\left[\begin{array}{cc}{b}_{11}& b_{12}\\ b_{21}& b_{22}\end{array}\right]u\right\}---\left(23\right)$

Therefore, the control law of design con-trol system is:

$u=-{\left[\begin{array}{cc}{b}_{11}& b_{12}\\ b_{21}& b_{22}\end{array}\right]}^{-1}\left[\begin{array}{c}{f}_V-\stackrel{\&CenterDot;\&CenterDot;\&CenterDot;}{V}+{\mathrm{\&beta;}}_{1}q/p{\stackrel{\&CenterDot;\&CenterDot;}{e}}_V^{\frac{q-1}{p}}+{\mathrm{\&beta;}}_2{q}_2/p_2{{\stackrel{\&CenterDot;}{e}}_V}^{\frac{q_1-1}{p_1}}\\ f_h-{h^{\left(4\right)}}_d+{\mathrm{\&alpha;}}_{1}q/p{{\stackrel{\&CenterDot;\&CenterDot;\&CenterDot;}{e}}_h}^{\frac{q-1}{p}}+{\mathrm{\&alpha;}}_2{q}_2/p_2{{\stackrel{\&CenterDot;\&CenterDot;}{e}}_h}^{\frac{q_1-1}{p_1}}+{\mathrm{\&alpha;}}_3{q}_2/p_2{{\stackrel{\&CenterDot;}{e}}_h}^{\frac{q_2-1}{p_2}}\end{array}\right]-k\mathrm{sgn}\left(S\right)---\left(24\right)$

(24) are substituted into (23), can obtain

$\begin{array}{c}\stackrel{\&CenterDot;}{V}=S^T(-k\mathrm{sgn}(S\left)\right)\\ \&le;-k||S||\end{array}$

Therefore system Existence of Global Stable.

Although control law given above can keep stablizing of system, but is because the problem that sliding-mode control designs, system would be likely to occur the buffeting of high frequency, here considers by taking the method for saturation function to suppress system chatter. System control law (24) is rewritten as:

$u=-{\left[\begin{array}{cc}{b}_{11}& b_{12}\\ b_{21}& b_{22}\end{array}\right]}^{-1}\left[\begin{array}{c}{f}_V-\stackrel{\&CenterDot;\&CenterDot;\&CenterDot;}{V}+{\mathrm{\&beta;}}_{1}q/p{\stackrel{\&CenterDot;\&CenterDot;}{e}}_V^{\frac{q-1}{p}}+{\mathrm{\&beta;}}_2{q}_2/p_2{{\stackrel{\&CenterDot;}{e}}_V}^{\frac{q_1-1}{p_1}}\\ f_h-{h^{\left(4\right)}}_d+{\mathrm{\&alpha;}}_{1}q/p{{\stackrel{\&CenterDot;\&CenterDot;\&CenterDot;}{e}}_h}^{\frac{q-1}{p}}+{\mathrm{\&alpha;}}_2{q}_2/p_2{{\stackrel{\&CenterDot;\&CenterDot;}{e}}_h}^{\frac{q_1-1}{p_1}}+{\mathrm{\&alpha;}}_3{q}_2/p_2{{\stackrel{\&CenterDot;}{e}}_h}^{\frac{q_2-1}{p_2}}\end{array}\right]-k\frac{S}{||S||+\mathrm{\&delta;}}---\left(25\right)$

Wherein �� is a little positive number, and such system is buffeted effect when close to sliding-mode surface and will be weakened.

Detailed description of the invention

In order to verify the effect of designed controller, and realizing effectively contrast, following by the data adopting document to provide and unit, the Cruise Conditions carrying out simulating, verifying aircraft is as shown in table 1:

In simulations, in formula (25), the value of parameter is as shown in table 2 below. Assume that given system instruction signals is: V_dT ()=15160ft/s, is namely that speed increases 100ft/s, investigates the performance of designed controller.

Comparing in many documents of reference, the system convergence time is 30s and more than 30s, and our convergence time is less than 20s, and this sliding-mode surface at Finite-time convergence, and can improve the tracking speed of system. Simulation result shows, given method for designing is feasible effectively.

Assume that given system instruction signals is: h_dT ()=112000ft, is namely highly increase 2000ft, investigate the performance of designed controller.

The Cruise Conditions of table 1 aircraft

The sliding-mode surface parameter value of table 2 design

Parameter	Value
		��1	4
��2	10
		��1	4
��2	4
		��3	10
q0/p0	11/13
		q1/p1	9/13
q2/p2	7/13

This method launches research for a quasi-nonlinear hypersonic vehicle, analyzing on the basis of vehicle dynamics model, the method adopting I/O linearization will be modeled as pro forma linear model, adopting Terminal sliding mode controller design method to devise output-tracked controller afterwards, designed controller ensure that system is at Finite-time convergence. Simulation result shows, designed controller is feasible effectively, it is possible to realize the rapid track and control to hypersonic aircraft.

Claims (1)

1. hypersonic aircraft Terminal sliding mode controller design method, it is characterised in that:

In single order Terminal sliding formwork, the definition of single order Terminal sliding formwork is as follows:

Wherein x is a scalar, �� > 0, p, q (p > q), and p, q are positive odd number; No matter x is any real number, x^p/qSolution be necessary for real number;

System dynamic property on sliding-mode surface is

Given arbitrary original state x (0) �� 0, then system will at Finite-time convergence to initial point; Solving equation (2)

The time that system experiences can be obtained from state x (0) to initial point

The explanation of Terminal limited mechanism is considered below, it is considered to the Jacobian matrix at equilibrium point x=0 place is

J is regarded as the eigenvalue �� of first order matrix, then when x �� 0⁺Time, J ��-��, then the track of system, naturally can with infinitely-great speed convergence to equilibrium point under the driving of the eigenvalue of minus infinity; Therefore system will at Finite-time convergence to initial point;

For nonlinear system, adopt the I/O linearization method based on differential geometric theory, the complexity reducing system controller design on the basis of mission nonlinear feature can retained, therefore it is contemplated that adopt I/O linearization method that system is processed;The party ratio juris is briefly described below;

For affine nonlinear system

Wherein f (x), g (x) are smooth function;

First passing through respectively to function f (x), g (x) asks for the Lie derivatives about output function h (x) system, seeks the Relative order of system, and concrete form is:

IfThe Relative order then claiming system is r;

If the Relative order r=n of system, (n is systematic education), then system is to fully input linearization; Selection differomorphism converts

To (8) formula derivation, the state equation that can obtain transformation system is:

Observation equation (4), it is possible to find except finally with respect to ��_nEquation outside, all the other n-1 equations have been linear forms, and without controlled quentity controlled variable; Only with respect to ��_nState equation be nonlinear, but to input u, equation form is linear; Dirty [state] equation (9)

Wherein

Then state equation can become:

WhereinAt this moment, system form becomes linear, and remain the nonlinear characteristic of system so that system more easily processes;

This method adopts the hypersonic aircraft longitudinal direction model that NASA langley laboratory is announced to study, and its model is as follows:

Model hypothesis

(1) hypersonic aircraft is desirable rigid body, is namely left out the elastic free degree of wing etc.;

(2) centroid position, rotary inertia is the function of quality, and centroid position changes at the axis longitudinal axis all the time;

(3) aircraft center and reference centre of moment are on body X-axis;

(4) assume that aircraft layout is symmetrical, namely product of inertia I_xy, I_xz, I_yzPerseverance is zero;

(5) rotary inertia and the motor power established angle of control surface are ignored;

In formula, L is lift, and D is resistance, T motor power, and Myy is rolling moment, and Iyy is aircraft own rotation inertia, and r is the distance of aircraft and the earth's core, and the expression formula that each parameter is concrete is as follows:

In formula, �� is atmospheric density, and S is that electromotor effective cross section is amassed, and the expression formula of each coefficient is as follows:

�� in formula_eRepresenting elevator drift angle, �� represents electromotor mode, and its expression formula is:

��_cInput is controlled for engine throttle;

Reality according to hypersonic aircraft, controls input �� by engine throttle_cWith elevator drift angle ��_eAs controlling input, speed V and height H is elected in output as;

According to (13)-(16) formula it is found that the kinetic model of hypersonic aircraft exists serious non-linear and close coupling, and not aobvious containing input in equation, it is impossible to directly design con-trol device, it is necessary to model is converted; I/O linearization method is the important method of Nonlinear Control System Design and process, make system form shows linear behavio(u)r by system is differentiated, and itself also retain the nonlinear characteristic of this system, therefore here adopt I/O linearization method to be converted by model, be then controlled device design;

Select state variable x=[V �� �� �� h]^T, control input u=[��_c��_e]^T, definition system is output as y=[V, h]^T, according to I/O linearization method, respectively to speed V with highly carry out 3 times and 4 subdifferentials, it is possible to obtain:

WhereinFor the ease of isolating controlled quentity controlled variable, selectWherein After differential, controlling input and be already present in the differential equation, output kinetics equation can be written as:

In formula

��₂=[��₂₁��₂₂��₂₃��₂₄��₂₅]

��₂=[��₂₁��₂₂��₂₃��₂₄��₂₅]

Converting through I/O linearization, the nonlinear model of hypersonic aircraft has been converted into pro forma linear model, and the nonlinear characteristic of model itself have also been obtained maximum reservation simultaneously;

The control target of hypersonic aircraft is to control one given command signal of aircraft output tracking, and ensures stablizing of system itself;Given command signal is y_com=[V_d(t),h_d(t)]^T, then following the tracks of target can be expressed as:The tracking error of definition system is:The target then controlled is the tracking error of guarantee systemBut the flight speed of hypersonic aircraft is fast, the relative response time requirement to system is also high, the controller of traditional exponential convergence is difficult to the requirement meeting aircraft to controlling system finite time convergence control, therefore the controller controlling to design finite time convergence control with Terminal sliding formwork is here considered, it is ensured that the Fast Convergent of system;

According to the thinking that Terminal sliding formwork controls, the sliding-mode surface of design tracking error is as follows:

HereIn like manner can push awayWithDesigned sliding-mode surfaceQi and pi (i=0,1,2) is positive odd number, and pi > qi; ��₁, ��₂, ��₁, ��₂And ��₃For arithmetic number;

According to the condition that sliding formwork controls, when system arrives sliding-mode surface, S=0, namely

Observation type (22) it is found thatIt is an equilibrium point of equation (22), and on sliding-mode surface, system equation convergence rate is power function convergence, so system can at Finite-time convergence to initial point; AndWhen, hypersonic aircraft also completes the effective tracking to command signal; So the sliding-mode surface of design meets the requirement of finite time convergence control;

Below for the sliding-mode surface of design, design corresponding sliding formwork control law, to ensure reachability and the global stability of System with Sliding Mode Controller;

Definition Lyapunov function is:

Above formula is differentiated:

Comparison formula (20), can obtain

Therefore, the control law of design con-trol system is:

(24) are substituted into (23), can obtain

Therefore system Existence of Global Stable;

Although control law given above can keep stablizing of system, but is because the problem that sliding-mode control designs, system would be likely to occur the buffeting of high frequency, here considers by taking the method for saturation function to suppress system chatter; System control law (24) is rewritten as:

Wherein �� is a little positive number, and such system is buffeted effect when close to sliding-mode surface and will be weakened.

C_D=resistance coefficient

C_L=lift coefficient

C_M(q)=inclination angle speed moment coefficient

C_M(��)=angle of attack moment coefficient

C_M(��_eThe inclined moment coefficient of)=rudder

C_T=thrust coefficient

=mean aerodynamic chord

D=resistance

H=height

I_yy=rotary inertia

L=lift

M_yy=pitching moment

M=mass

Q=inclination angle speed

R_E=earth radius

R=is from geocentric distance

S=reference area

T=thrust

V=speed

��=angle of attack

��=throttle valve control

��=flight-path angle

��_e=rudder deviator

��=gravity coefficient

��=densityofair.